Math 425, Spring 2015 Jerry L. Kazdan Problem Set 1 Due: Thurs. Jan. 22 in class. [Late papers will be accepted until 1:00 PM Friday.] This is rust remover. It is essentially Homework Set 0 with a few modifications. Notation: u t = u t. This week. Please read all of Chapter 1 in the Strauss text. 1. Let u(t) be the solution of u = 3u with initial value u(0) = A > 0. At what time T is u(t) = 2A? 2. Let u(t) be the amount of a radioactive element at time t and say initially, u(0) = A > 0. The rate of decay is proportional to the amount present, so du dt = cu, where the constant c > 0 determines the decay rate. The half-life T is the amount of time for half of the element to decay, so u(t) = 1 2u(0). Find c in terms of T and obtain a formula for u(t) in terms of T. 3. Let x 0 constant A. f(t)dt = e cos(3x) + A, where f is some continuous function. Find f and the 4. a) If u + 4u = 0 with initial conditions u(0) = 1 and u (0) = 2, compute u(t). b) Find a particular solution of the inhomogeneous equation u + 4u = 8. c) Find a particular solution of the inhomogeneous equation u + 4u = 4t. d) Find a particular solution of the inhomogeneous equation u + 4u = 8 8t. e) Find the most general solution of the inhomogeneous equation u + 4u = 8 8t. f) If f(t) is any continuous function, use the method variation of parameters (look it up if you don t know it) to find a formula for a particular solution of u +4u = f(t). 5. Let u(t) be any solution of u + 2bu + 4u = 0. If b > 0 is a constant, show that lim t u(t) = 0. 6. a) If u 4u = 0 with initial conditions u(0) = 1 and u (0) = 2, compute u(t). b) Find a particular solution of the inhomogeneous equation u 4u = 8. c) Find a particular solution of the inhomogeneous equation u 4u = 4t. 1
d) Find a particular solution of the inhomogeneous equation u 4u = 8 8t. e) Find the most general solution of the inhomogeneous equation u 4u = 8 8t. f) If f(t) is any continuous function, use the method variation of parameters (look it up if you don t know it) to find a formula for a particular solution of u 4u = f(t). 7. Say w(t) satisfies the differential equation aw (t) + bw + cw(t) = 0, (1) where a and c, are positive constants and b 0. Let E(t) = 1 2 [aw 2 + cw 2 ]. a) Without solving the differential equation, show that E (t) 0. b) Use this to show that If you also know that w(0) = 0 and w (0) = 0, then w(t) = 0 for all t 0. c) [Uniqueness] Say the functions u(t) and v(t) both satisfy the same equation (1) and also u(0) = v(0) and u (0) = v (0). Show that u(t) = v(t) for all t 0. 8. Say u(x, t) has the property that u t = 2 for all points (x, t) R2. a) Find some function u(x, t) with this property.. b) Find the most general such function u(x, t). c) If u(x, 0) = sin 3x, find u(x, t). d) If instead u satisfies u = 2xt, still with u(x,0) = sin3x, find u(x, t). t 9. Say u(x, t) has the property that u t = 3u for all points (x, t) R2. a) Find some such function other than the trivial u(x, t) 0. b) Find the most general such function. c) If u(x, t) also satisfies the initial condition u(x,0) = sin3x, find u(x, t). 10. a) If u(x, t) = cos(x 3t) + 2(x 3t) 7, show that 3u x + u t = 0. b) If f(s) is any smooth function of s and u(x, t) = f(x 3t), show that 3u x +u t = 0. 11. A function u(x, y) satisfies 3u x + u t = f(x, t), where f is some specified function. a) Find an invertible linear change of variables r =ax + bt s =cx + dt, 2
where a, b, c, d are constants, so that in the new (r, s) variables u satisfies u s = g(r, s), where g is related to f by the change of variables. [Remark: There are many possible such changes of variable. The point is to reduce the differential operator 3u x + u t to the much simpler u s.] b) Use this procedure to solve 3u x + u t = 1 + x + 2t with u(x,0) = e x. 12. Let S and T be linear spaces, such as R 3 and R 7 and L : S T be a linear map; thus, for any vectors X, Y in S and any scalar c L(X + Y ) = LX + LY and L(cX) = cl(x). Say V 1 and V 2 are (distinct!) solutions of the equation LX = Y 1 while W is a solution of LX = Y 2. Answer the following in terms of V 1, V 2, and W. a) Find some solution of LX = 2Y 1 7Y 2. b) Find another solution (other than W) of LX = Y 2. 13. The following is a table of inner ( dot ) products of vectors u, v, and w. For example, v w = w v = 3. u v w u 4 0 8 v 0 1 3 w 8 3 50 a) Find a unit vector in the same direction as u. b) Compute u (v + w). c) Compute v + w. d) Find the orthogonal projection of w into the plane E spanned by u and v. [Express your solution as linear combinations of u and v.] e) Find a unit vector orthogonal to the plane E. f) Find an orthonormal basis of the three dimensional space spanned by u, v, and w. 14. Let z and w be complex numbers. a) Write the complex number z = 1 3 + 4i real numbers. in the form z = a + ib where a and b are 3
b) Show that (zw) = z w. c) Show that z 2 = z z. d) show that zw = z w. 15. If z = x + iy is a complex number, one way to define e z is by the power series e z = 1 + z + z2 2! + z3 3! + + zk k! + = z n n!. (2) n=0 a) Using the usual (real) power series for cosy and siny, show that e iy = cos y + i siny. b) Use this to show that cos y = eiy + e iy and siny = eiy e iy. 2 2i c) Using equation (2), one can show that e z+w = e z e w for any complex numbers z and w (accept this for now). Consequently e i(x+y) = e ix e iy. Use the result of part (a) to show that this implies the usual formulas for cos(x+y) and sin(x + y). 16. Let D R 2 be a bounded (connected) region with smooth boundary B. If u(x, y) is a smooth function, write u = u xx + u yy (we call the Laplace operator). Some people write u = 2 u. Suggestion: First do this problem for a function of one variable, u(x), so u = u and, say, D is the interval {0 < x < 1}. a) Show that u u = (u u) u 2. b) If u(x, y) = 0 on B. Show that u u dx dy = D D u 2 dx dy. c) If u = 0 in D and u = 0 on the boundary B, show that u(x, y) = 0 throughout D. 17. The temperature u(x, t) of a certain thin rod, 0 x L satisfies the heat equation u t = u xx (3) 4
Assume the initial temperature u(x,0) = 0 and that both ends of the rod are kept at a temperature of 0, so u(0, t) = u(l, t) = 0 for all t 0. What do you anticipate the temperature in the rod will be at any later time t? I hope you suspect that u(x, t) = 0 for all t 0. Use the following to prove this. Let H(t) = L 0 u 2 (x, t)dx. a) Show that since the temperature on the ends of the rod is always zero, then dh/dt 0 (an integration by parts will be needed). Thus, for any t 0 we know that H(t) H(0). b) Since the initial temperature is zero, what is H(0)? Why does this imply that H(t) = 0 for all t 0? Why does this imply that u(x, t) = 0 for all points on the rod and all t 0? c) [Uniqueness] Say that the functionns u(x, t) and v(x, t) both satisfy the heat equation (3) and have the identical initial values and boundary values: u(x,0) = v(x,0) for 0 x L, u(x, t) = v(x, t) for x = 0 and x = L, t 0. Show that u(x, t) = v(x, t) for all 0 x L, t 0. 18. [Generalization of Problem 17 to more space dimensions]. Say a function u(x, y, t) satisfies the heat equation in a bounded region Ω R 2 : u t = u xx + u yy (4) and that u(x, y, t) = 0 for all points (x, y) on the boundary, B of Ω. Similar to Problem 17, define H(t) = u 2 (x, y, t)dx dy. a) Show that dh/dt 0. [Suggestion: See Problem 16.] Ω b) If in addition you know that the initial temperature is zero, u(x, y,0) = 0 for all points (x, y) Ω, show that u(x, y, t) = 0 for all (x, y) Ω and all t 0. c) [Uniqueness] Say that the functionns u(x, y, t) and v(x, y, t) both satisfy the heat equation (4) and have the identical initial values and boundary values: u(x, y,0) = v(x, y,0) for (x, y) Ω, u(x, y, t) = v(x, y, t) for (x, y) on B, and t 0. Show that u(x, y, t) = v(x, y, t) for all (x, y) Ω t 0. [Last revised: January 23, 2015] 5